Step 1 :The problem provides the lengths of the sides of a rectangular cage as \((3x - 1)\) ft and \((-4x + 1)\) ft.
Step 2 :We are asked to find an expression for the area of the cage.
Step 3 :The area of a rectangle is given by the product of its length and width.
Step 4 :So, we multiply the expressions for the length and width to find the area: \((3x - 1) * (-4x + 1)\).
Step 5 :Solving this, we get the expression for the area as \(-12x^2 + 7x - 1\).
Step 6 :Thus, the correct expression for the area of the rectangular cage is \(\boxed{-12x^2 + 7x - 1}\).